Geodesic
DIVISION SUDOKUS: INVARIANTS, ENUMERATION, AND MULTIPLE PARTITIONS
Glasgow mathematical journal, Tome 62 (2020) no. 3, pp. 600-630

Voir la notice de l'article provenant de la source Cambridge University Press

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions.
DOI : 10.1017/S0017089519000375
Mots-clés : 05B15, 20N05 
DRÁPAL, ALEŠ; VOJTĚCHOVSKÝ, PETR. DIVISION SUDOKUS: INVARIANTS, ENUMERATION, AND MULTIPLE PARTITIONS. Glasgow mathematical journal, Tome 62 (2020) no. 3, pp. 600-630. doi: 10.1017/S0017089519000375
@article{10_1017_S0017089519000375,
     author = {DR\'APAL, ALE\v{S} and VOJT\v{E}CHOVSK\'Y, PETR},
     title = {DIVISION {SUDOKUS:} {INVARIANTS,} {ENUMERATION,} {AND} {MULTIPLE} {PARTITIONS}},
     journal = {Glasgow mathematical journal},
     pages = {600--630},
     year = {2020},
     volume = {62},
     number = {3},
     doi = {10.1017/S0017089519000375},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000375/}
}
TY  - JOUR
AU  - DRÁPAL, ALEŠ
AU  - VOJTĚCHOVSKÝ, PETR
TI  - DIVISION SUDOKUS: INVARIANTS, ENUMERATION, AND MULTIPLE PARTITIONS
JO  - Glasgow mathematical journal
PY  - 2020
SP  - 600
EP  - 630
VL  - 62
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000375/
DO  - 10.1017/S0017089519000375
ID  - 10_1017_S0017089519000375
ER  - 
%0 Journal Article
%A DRÁPAL, ALEŠ
%A VOJTĚCHOVSKÝ, PETR
%T DIVISION SUDOKUS: INVARIANTS, ENUMERATION, AND MULTIPLE PARTITIONS
%J Glasgow mathematical journal
%D 2020
%P 600-630
%V 62
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000375/
%R 10.1017/S0017089519000375
%F 10_1017_S0017089519000375

[1] 3Doku, https://www.tellini.org/products/mac/3doku. Google Scholar

[2] Bailey, R. A., Cameron, P. J. and Connelly, R., Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes, Amer. Math. Month. 115(5) (2008), 383–404. Google Scholar | DOI

[3] Belousov, V. D., The group associated with a quasigroup (Russian), Mat. Issled. 4/3 (1969), 21–39. Google Scholar

[4] Bruck, R. H., A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 20 (Springer Verlag, Berlin-Göttingen-Heidelberg, 1958). Google Scholar | DOI

[5] Cameron, P. J., Hilton, A. J. W. and Vaughan, E. R., An analogue of Ryser’s theorem for partial Sudoku squares, J. Combin. Math. Combin. Comput . 80 (2012), 47–69. Google Scholar

[6] Drápal, A. and Lisoně, P., Extreme nonassociativity via nearfields, accepted to Finite Fields and Applications. Google Scholar

[7] Drápal, A. and Valent, V., High nonassociativity in order 8 and an associative index estimate, J. Combin. Des . 27 (2019), 205–228. Google Scholar

[8] Drápal, A. and Valent, V., Extreme nonassociativity in order nine and beyond, J. Combin. Des., accepted. Google Scholar

[9] Colbourn, C. J. and Dinitz, J. H. (eds.), Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton), 2nd edition (Chapman and Hall/CRC, Boca Raton, FL, 2007). Google Scholar

[10] The, Gap Group, GAP – Groups, Algorithm s, and Programming, Version 4.6.3 (2013). http://www.gap-system.org. Google Scholar

[11] Keedwell, A. D., Constructions of complete sets of orthogonal diagonal Sudoku squares, Australas. J. Combin. 47 (2010), 227–238. Google Scholar

[12] Kepka, T., A note on associative triples of elements in cancellation groupoids, Comment. Math. Univ. Carolin. 21 (1980), 479–487. Google Scholar

[13] Lambert, T. A. and Whitlock, P. A., Generalizing Sudoku to three dimensions, Monte Carlo Methods Appl . 16(3–4) (2010), 251–263. Google Scholar | DOI

[14] Lorch, J., Magic squares and sudoku, Amer. Math. Month. 119(9) (2012), 759–770. Google Scholar | DOI

[15] Mccune, W., Mace4, https://www.cs.unm.edu/mccune/mace4. Google Scholar

[16] Mcguire, G., Tugemann, B. and Civario, G., There is no 16-clue Sudoku: solving the Sudoku minimum number of clues problem via hitting set enumeration, Exp. Math. 23(2) (2014), 190–217. Google Scholar | DOI

[17] Nagy, G. P. and Vojtěchovský, P., LOOPS: Computing with quasigroups and loops in GAP. Available at http://www.math.du.edu/~petr/loops. Google Scholar

[18] Pedersen, R. M. and Vis, T. L., Sets of mutually orthogonal Sudoku Latin squares, College Math. J. 40(3) (2009), 174–180. Google Scholar

[19] Russell, E. and Jarvis, F., http://www.afjarvis.staff.shef.ac.uk/sudoku/sudgroup.html. Google Scholar

[20] Stanovský, D. and Vojtěchovský, P., Commutator theory for loops, J. Algebra 399 (2014), 290–322. Google Scholar | DOI

[21] Stein, S. K., Homogeneous quasigroups, Pacific J. Math . 14 (1964), 1091–1102. Google Scholar | DOI

[22] Su-Doku’s maths, http://forum.enjoysudoku.com/su-doku-s-maths-t44.html. Google Scholar

[23] Zassenhaus, H., Über endliche Fastkörper (German), Abh. Math. Sem. Univ. Hamburg 11(1) (1935) 187–220. Google Scholar | DOI

Cité par Sources :